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Parity effects in the scaling of block entanglement in gapless spin chains
Calabrese, P.; Campostrini, M.; Essler, F.; Nienhuis, B.
DOI
10.1103/PhysRevLett.104.095701
Publication date
2010
Document Version
Final published version
Published in
Physical Review Letters
Link to publication
Citation for published version (APA):
Calabrese, P., Campostrini, M., Essler, F., & Nienhuis, B. (2010). Parity effects in the scaling
of block entanglement in gapless spin chains. Physical Review Letters, 104(9), 095701.
https://doi.org/10.1103/PhysRevLett.104.095701
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Parity Effects in the Scaling of Block Entanglement in Gapless Spin Chains
Pasquale Calabrese,1Massimo Campostrini,1Fabian Essler,2and Bernard Nienhuis3
1Dipartimento di Fisica, dell’Universita` di Pisa and INFN, 56127 Pisa, Italy
2The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom 3
Institute for Theoretical Physics, Universiteit van Amsterdam, 1018 XE Amsterdam, The Netherlands (Received 1 December 2009; revised manuscript received 21 January 2010; published 5 March 2010) We consider the Re´nyi entropies for Luttinger liquids (LL). For large block lengths ‘, these are known to grow like ln‘. We show that there are subleading terms that oscillate with frequency 2kF(the
Fermi wave number of the LL) and exhibit a universal power-law decay with ‘. The new critical exponent is equal to K=ð2Þ, where K is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.
DOI:10.1103/PhysRevLett.104.095701 PACS numbers: 64.70.Tg, 03.67.Mn, 05.70.Jk, 75.10.Pq
Luttinger liquid (LL) theory describes the low-energy (large-distance) physics of gapless one-dimensional mod-els such as quantum spin chains and correlated electron models. It corresponds to a conformal field theory (CFT) with central charge c ¼ 1 and is known to provide accurate predictions for universal properties of many physical sys-tems. LL theory has been applied successfully to recent experiments on carbon nanotubes [1], spin chains [2], and cold atomic gases [3]. A much studied example of a lattice model that gives rise to a LL description at low energies is the spin-1=2 Heisenberg XXZ chain
H ¼ X L j¼1 ½x jxjþ1þ y j y jþ1þ zjzjþ1: (1)
Here, jare Pauli matrices at site j, and we have imposed
periodic boundary conditions. Recent years have witnessed a significant effort to quantify the degree of entanglement in many-body systems (see, e.g., [4] for reviews). Among the various measures, the entanglement entropy (EE) has been by far the most studied. By partitioning an extended quantum system into two subsystems, the EE is defined as the von Neumann entropy of the reduced density matrix A
of one of the subsystems. The leading contribution to the EE of a single, large block of length ‘ can be derived by general CFT methods [5–7]. The case of a subsystem consisting of multiple blocks requires a model dependent treatment, but the EE can still be obtained from CFT [8]. On the other hand, little is known with regard to correc-tions to the leading asymptotic behavior. In the following, we consider the Re´nyi entropies
S¼
1
1 lnTr
A; (2)
which give the full spectrum of A[9] and are fundamental
for understanding the scaling of algorithms based on ma-trix product states [10–12]. We note that S1 is the
von Neumann entropy and S1 gives minus the logarithm
of the maximum eigenvalue of the reduced density matrix
(known as single copy entanglement [13,14]). According to CFT, in an infinite gapless one-dimensional model, a block of length ‘ has entropies [5–7]
SCFT ð‘Þ ¼ c 6 1 þ1 ln‘ þ c0; (3)
where c is the central charge and c0 a nonuniversal
con-stant. In a finite system of length L, the block length ‘ in (3) should be replaced with the chord distance Dð‘; LÞ ¼
L
sin‘L . In many lattice models, the asymptotic scaling is
obscured by large oscillations proportional toð1Þ‘. Some typical examples are shown in Fig. 1, where we plot Sð‘; LÞ for ¼ 1; 2; 1 for the XXZ model at ¼ 1=2
as obtained by density matrix renormalization group (DMRG) computations. While S1 is smooth, SÞ1 is seen
to exhibit large oscillations. For ¼ 1, in particular, it is difficult to recognize the CFT scaling behavior (3). While such oscillations have been observed in several examples [15,16] and can be seen to arise from strong antiferromag-netic correlations, a quantitative understanding of these features was until now lacking. We show that these
oscil-FIG. 1 (color online). Parity effects in Re´nyi entropies in the XXZ model at ¼ 1=2. Sð‘; LÞ for several L and ¼
1; 2; 1. The straight lines indicate the asymptotic slopes ð1 þ 1Þ=6.
lations obey the universal scaling law
Sð‘Þ SCFT ð‘Þ ¼ fcosð2kF‘Þj2‘ sinkFjp; (4)
where p is a universal critical exponent equal to 2K=.
Here, K is the LL parameter, kF is the Fermi momentum,
and f is a nonuniversal constant. In a finite system, the
block length ‘ in (4) is replaced by the chord distance, and fis multiplied by a universal scaling function Fð‘=LÞ,
that in general depends on the parity of L. We note that in zero magnetic field (half-filling), we have kF ¼ =2, and
the oscillating factor in (4) reduces to ð1Þ‘ as observed.
While we establish (4) for the particular case of the Heisenberg XXZ chain (1), where the LL parameter is given by K ¼ =ð2 arccosÞ, we expect the scaling form to be universal because it is related to the low-energy excitations of the model and is therefore encoded in the continuum LL field theory description. Recent results for the entanglement entropy confirm these expectations [17]. XX model.—This case corresponds to ¼ 0 in (1). The LL parameter and exponent in (4) are K ¼ 1 and p¼
2=, respectively. The computation of the Re´nyi entropies can be achieved by exploiting the Jordan-Wigner mapping to free fermions, which reduces the problem to the diago-nalization of an ‘ ‘ correlation matrix (see [18] for details). Jin and Korepin (JK) showed [19] that Re´nyi entropies can be obtained by the following contour integral encircling the segment½1; 1 of the real axis
Sð‘Þ ¼ 1 2i I eðÞ d lnD‘ðÞ d d: (5) Here, D‘ðÞ is the determinant of a ‘ ‘ Toeplitz matrix
and eðÞ ¼11 ln½ð1þ2 Þþ ð12 Þ. In [19], the
Fisher-Hartwig formula was used to determine the asymptotic scaling of the Re´nyi entropy with ‘, which agrees with the CFT formula (3). Here, we employ the generalized Fisher-Hartwig conjecture [20] in order to go beyond the results of [19] and determine the subleading corrections. The terms in the asymptotic expansion of the determinant relevant for calculation of the Re´nyi entropy can be cast in the form D‘ðÞ DJK ‘ ðÞ ¼ 1 þ e2ikF‘L2½1þ2ðÞ k 2ð1 þ ðÞÞ 2ð ðÞÞ þ e2ikF‘L2½12ðÞ k 2ð1 ðÞÞ 2ððÞÞ ; (6) where DJK
‘ is the result of [19], 2iðxÞ ¼ ln½ð1 þ
xÞ=ð1 xÞ, and Lk¼ j2‘ sinkFj. The calculation of the
integral in (5) is now straightforward. One expands lnD‘in
(6) in powers of Lk, determines the discontinuity across the
cut ½1; 1, changes the integration variable from to iðÞ, and finally obtains the leading behavior from the poles closest to the real axis (details will be reported elsewhere [21]). The resulting asymptotic expression is valid at fixed , in the limit lnLk . The final result is
given by Eq. (4) with
f¼
2 1
ðð1 þ Þ=2Þ
2ðð1 1Þ=2Þ: (7) We note that f1 ¼ 0, and therefore no oscillating
correc-tions for the Von-Neumann entropy are predicted, in agree-ment with numerical observations.
The requirement that lnLk implies that the
asymp-totic behavior is only reached for very large block lengths, e.g., at ¼ 10, we need Lk 20 000. In the
preasymp-totic regime, there are several sources of corrections. First, the integral is no longer dominated by the poles closest to the real axis, which leads to power-law corrections of the form L2m=k (with integer m), which oscillate as ei2kF‘.
Corrections with different oscillatory behavior arise from the higher order terms in the expansion of lnD‘ðÞ in
powers of Lk. The first correction is proportional to
ei4kF‘, the next to ei6kF‘, etc. In zero magnetic field,
where kF ¼ =2, the leading term is proportional to ð1Þ‘
while the second does not oscillate. Hence, there is a subleading constant background in addition to the leading oscillatory behavior. In the limit ! 1, all these terms become of the same order so that we need to resum the entire series that arises from expanding lnD‘ðÞ and then
carrying out the integral. In the zero magnetic field case, we thus obtain S1ð‘Þ SJK1ð‘Þ ¼ (2 12 lnbL1 k ‘odd; 2 24 lnbL1 k ‘even: (8)
Here, the constant b 7:1 has been fixed by summing certain contributions to all orders in 1=ðlnLkÞ and agrees
well with numerical results [22].
Numerical results for the XX model can be obtained by diagonalizing the correlation matrix both infinite and finite systems. We first present the results for infinite systems. We consider only the model in zero magnetic field and plot the quantity
dð‘Þ Sð‘Þ SCFT ð‘Þ (9)
where the value for the constant contribution c0in SCFT ð‘Þ
is taken from [19]. According to Eq. (4) for kF ¼ =2,
dð‘Þ ’ ðÞ‘ð2‘Þpf. In Fig.2, we compare the
abso-lute value of dð‘Þ for ¼ 2; 5; 20; 1 and block sizes ‘ up
to 4000 sites to our asymptotic results (4) and (8). For ¼ 2, the curves for odd and even ‘ are practically indistin-guishable (the line corresponding to the analytical result is invisible under the data points). For ¼ 5, we still obtain power laws with the exponent p5¼ 2=5, but the curves are
not as symmetrical as for ¼ 2 because subleading cor-rections become visible. Increasing further, the devia-tions of dð‘Þ for ‘ < 4000 from the asymptotic behavior
become quite pronounced. This is shown in Fig.2for the case ¼ 20, where the leading asymptotic result (straight line) is seen to be a poor approximation to dð‘Þ for even ‘.
Including the subleading corrections gives curves perfectly covered by data in Fig.2. The last panel in Fig.2shows the 095701-2
result in the ¼ 1. The numerical results are seen to be in perfect agreement with Eq. (8).
We now turn to finite systems. We numerically deter-mined the quantity [recall that Dð‘; LÞ ¼L
sin‘L ]
Fð‘=LÞ ¼ ½Sð‘; LÞ SCFT ð‘; LÞf1 Dð‘; LÞ2=; (10)
[for ¼ 1, we multiply by lnbDð‘; LÞ] for a variety of values of and system sizes ranging from L ¼ 17 to L ¼ 4623. We observe that there is data collapse for any L on two scaling functions for ‘ odd and even, respectively. Results for the cases ¼ 2; 1 and odd L are shown in Fig.3. The quality of the collapse is impressive considering that there are no adjustable parameters and that the plots contain millions of points ranging over 3 orders of magni-tude in both ‘ and L. For ¼ 2, we observe that F2ðxÞ ¼
cosx (these are shown as continuous curves in Fig.3). We currently have no analytical derivation of this scaling function. For other values of , we obtain similar data collapse, but the quality decreases with increasing , in-dicating the presence of other corrections. For even L, we obtain different scaling functions—FðxÞ then is almost
constant (see below).
XXZ model and DMRG.—To characterize the XXZ model in the gapless phase with 1 < 1, we per-formed extensive DMRG calculations at finite L. We used the finite-volume algorithm keeping ¼ 800 states in the decimation procedure. This rather large value of is required to obtain a precise determination of the full spec-trum of the reduced density matrix in the case of periodic boundary conditions. The data we have used in our analysis can be considered as numerically exact. Hence, the main limitation as compared to the XX case is the relatively small value of L accessible by DMRG (we considered 21 L 81 for odd L and 20 L 80 for even L). Another complication stems from the fact that the value of the constant contribution c0to the Re´nyi entropy is not
known analytically for Þ 0 [23]. We obtain it by fitting
our numerical data. The results are shown in Fig. 4. The data for ¼ 0 is in good agreement with the exact results of [19] (full line), establishing the correctness of our fitting procedure and the reliability of DMRG. The multiplicative constant c¼ eð1Þc
0
in the moments of the density
matrix TrA ¼ ca‘c=6ð1=Þ in Fig.4shows an
exponen-tial decay with , except for very close to 1. Hence, c
can essentially be absorbed into a rescaled block length ‘ as was pointed out in [9,24].
Having obtained the constant contribution c0, one can
determine the universal scaling functions FðxÞ. We
present results for both even and odd L and a number of representative values of and in Fig.5. We find that the data collapse is very good for all cases. This is remarkable given the limited system sizes accessible by DMRG. We note that, as expected, the data collapse becomes poor in the vicinity of the two isotropic points ¼ 1. At ¼ 1, there is a marginal operator (see, e.g., [15]) that gives rise to well-known logarithmic corrections to scaling for
FIG. 2 (color online). Corrections to scaling dð‘Þ ¼ Sð‘Þ
SCFT
ð‘Þ for the XX model and four different values of . Red
circles and black squares correspond to even and odd ‘, respec-tively.
FIG. 4 (color online). Left: The additive constant in the Re´nyi entropies c0(bottom) and the multiplicative constant cfor the
moments Tr
A(top in log scale) as function of at fixed . The
drawn lines are the exact values for ¼ 0. Right: c0as function
of at fixed .
FIG. 3 (color online). Corrections to scaling in the finite length XX model for odd L. F2ðxÞ and F1ðxÞ (x ¼ ‘=L) obtained from
69 different systems with lengths in the range 17 L 4623 exhibit perfect data collapse.
correlation functions. In the ferromagnetic limit ! 1, the model loses conformal invariance (the dispersion rela-tion becomes quadratic) and is no longer described by a LL. Hence, none of the results presented here is expected to hold.
The critical Ising chain has no strong antiferromagnetic correlations, and we therefore expect the corrections to scaling to be nonoscillatory. It is easy to see that this is indeed the case. Igloi and Juhasz [25] showed that the EEs of the XY chain can be expressed in terms of the EEs of two Ising chains. At the quantum critical point, this relation reads SXX
ð2‘; 2LÞ ¼ 2SIð‘; LÞ, where SI refers to the
critical Ising chain. This implies that the Re´nyi entropies in the Ising chain are just one half of the corresponding entropies in a XX chain of twice the block length and twice the system size. As both ‘ and L are even, our results for the XX model imply that the corrections to scaling are nonoscillatory and decay as ‘2=. This agrees with nu-merical computations.
Conclusions.—In this Letter, we considered the Re´nyi entropies for the critical spin-1=2 Heisenberg XXZ chain with periodic boundary conditions. By a combination of analytic and numerical techniques, we computed oscillat-ing corrections to scaloscillat-ing which are expected to be univer-sal. These are parametrized in terms of the Luttinger parameter K and the Fermi momentum kF. We argued
that our results hold generally for Luttinger liquids. In the case of open boundary conditions, a similar relation-ship is expected to hold, but with the replacement K ! K=2 (see also [15]). It would be interesting to prove this at least in the special case of free fermions. Finally, we would like to comment on our results in light of a recent proposal [8], that one way of distinguishing between different theo-ries with the same central charge is to consider the entan-glement of multiple blocks. Our results establish that it is
sufficient to consider a single block once one takes into account the universal subleading oscillatory corrections.
We thank J. Cardy for very helpful discussions. This work was supported by the ESF network INSTANS (P. C) and the EPSRC under Grant No. EP/D050952/1 (F. H. L. E.).
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